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The Structure of Crystalline Solids

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2019 Fall Introduction to Materials Science and Engineering 09. 19. 2019 Eun Soo Park Office: 33‐313 Telephone: 880‐7221 Email: [email protected] Office hours: by appointment 1 Contents for previous class CHAPTER 3: Fundamentals of Crystallography I. Crystal Structures - Lattice, Unit Cells, Crystal system II. Crystallographic Points, Directions, and Planes - Point coordinates, Crystallographic directions, Crystallographic planes III. Crystalline and Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy, Noncrystalline solids 2 Stacking of atoms in solid Finding stable position 2 1 • Minimize energy configuration – Related to the bonding nature 3 I. Crystal structure Lattice : 결정 공간상에서 점들의 규칙적인 기하학적 배열 – 3D point array in space, such that each point has identical surroundings. These points may or may not coincide with atom positions. – Simplest case : each atom its center of gravity point or space lattice pure mathematical concept example: sodium (Na) ; body centered cubic Hard‐sphere unit cell Reduced sphere unit cell Aggregate of many atoms I. Crystal structure : Unit cell 14 Bravais Lattice - Only 14 different types of unit cells are required to describe all lattices using symmetry rhombohedral cubic hexagonal tetragonal orthorhombic monoclinic triclinic (trigonal) P I F C 5 II. Crystallographic points, directions and planes Chapter 3.5 Point coordinates • position: fractional multiples of the unit cell edge lengths – ex) P: q,r,s cubic unit cell 6 Crystallographic Directions • a line between two points or a vector • [uvw] square bracket, smallest integer • families of directions: <uvw> angle bracket cubic 7 Chapter 3.7 Crystallographic Planes Lattice plane (Miller indices) z p m00, 0n0, 00p: define lattice plane m, n, ∞ : no intercepts with axes c Intercepts @ 213 n y (mnp) a b Reciprocals ½ 1 1/3 Miller indicies 3 6 2 m (362) plane x Plane (hkl) Family of planes {hkl} Miller indicies ; defined as the (362) smallest integral multiples of the reciprocals of the plane intercepts on the axes 8 Miller-Bravais vs. Miller index system Directions in Hexagonal system Conversion of 4 index system (Miller-Bravais) to 3 index (Miller) t u'a v'a w'c ua va ta wc Ex. M [100] 1 2 1 2 3 u=(1/3)(2*1-0)=2/3 v=(1/3)(2*0-1)=-1/3 Miller-Bravais to Miller Miller to Miller-Bravais w=0 4 to 3 axis 3axis to 4 axis system => 1/3[2 -1 -1 0] 1 u' u t 2u v u (2u'v') 3 Ex. M-B [1 0 -1 0] u’=2*1+0=2 v' v t 2v u 1 v (2v'u') v’=2*0+1=1 w' w 3 w’=0 => [2 1 0] w w' 9 Contents for today’s class CHAPTER 3: Fundamentals of Crystallography III. Crystalline ↔ Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy - Quasicrystals - Noncrystalline solids : Amorphous solid CHAPTER 4: The Structure of Crystalline Solids 10 III. Crystalline ↔ Noncrystalline Materials CRYSTALS AS BUILDING BLOCKS • Some engineering applications require single crystals: --diamond single crystals --turbine blades Natural and artificial Fig. 8.30(c), Callister 6e. (Fig. 8.30(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) • Crystal properties reveal features of atomic structure. --Ex: Certain crystal planes in quartz fracture more easily than others. (Courtesy P.M. Anderson) 11 Solidification: Liquid Solid 4 Fold Anisotropic Surface Energy/2 Fold Kinetics, Many Seeds 12 Grain Boundaries 13 Single vs Polycrystals • Single Crystals E (diagonal) = 273 GPa Data from Table 3.3, -Properties vary with Callister 7e. (Source of data is R.W. direction: anisotropic. Hertzberg, Deformation and Fracture Mechanics of -Example: the modulus Engineering Materials, 3rd ed., John Wiley and Sons, of elasticity (E) in BCC iron: 1989.) E (edge) = 125 GPa • Polycrystals -Properties may/may not 200 m Adapted from Fig. 4.14(b), Callister 7e. vary with direction. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, -If grains are randomly the National Bureau of Standards, Washington, oriented: isotropic. DC [now the National Institute of Standards and (Epoly iron = 210 GPa) Technology, Gaithersburg, -If grains are textured, MD].) anisotropic. 14 Polycrystals Anisotropic • Most engineering materials are polycrystals. Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm • Nb-Hf-W plate with an electron beam weld. Isotropic • Each "grain" is a single crystal. • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). 15 Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism) iron system titanium liquid , -Ti 1538ºC -Fe carbon BCC diamond, graphite 1394ºC FCC -Fe 912ºC BCC -Fe 16 DEMO: HEATING AND COOLING OF AN IRON WIRE The same atoms can • Demonstrates "polymorphism" have more than one crystal structure. Temperature, C Liquid 1536 BCC Stable 1391 longer heat up FCC Stable shorter! 914 BCC Stable longer! cool down Tc 768 magnet falls off shorter 17 Non-crystalline Materials Amorphous materials a wide diversity of materials can be rendered amorphous indeed almost all materials can. - metal, ceramic, polymer ex) amorphous metallic alloy (1960) - glassy/non-crystalline material cf) amorphous vs glass Amorphous - random atomic structure (short range order) - showing glass transition. Glass - retain liquid structure - rapid solidification from liquid state 18 Obsidian is a naturally occurring volcanic glass formed as an extrusive igneous rock. It is produced when felsic lava extruded from a volcano cools rapidly with minimum crystal growth. Obsidian is commonly found within the margins of rhyolitic lava flows known as obsidian flows, where the chemical composition (high silica content) induces a high viscosity and polymerization degree of the lava. The inhibition of atomic diffusion through this highly viscous and polymerized lava explains the lack of crystal growth. Because of this lack of crystal structure, abcde obsidian blade edges can reach almost molecular thinness, leading to its ancient use as projectile points and blades, and its modern use as surgical scalpel blades. Glass formation: stabilizing the liquid phase First metallic glass (Au80Si20) produced by splat quenching at Caltech by Pol Duwez in 1957. 1500 1414 -RapidRelative splat decrease quenching of 1300 Liquid mix meltingliquid metal temperature droplet Tm 1064.4 laser 1100 T mix T trigger T* m m 0.2 900 mix deep eutectic Tm metal anvil 700 ΔT* = 0.679 in most of glass forming alloys 500 mix i 363 (where, T m x iT m , 18.6 metal 300 xi = mole fraction,piston Tm i 100 Tm = melting point) 01020 30 40 50 60 70 80 90 100 Au Si t = 20 μm w = 2 cm W. Klement, R.H. Willens, P. Duwez, Nature 1960; 187: 869. by I.W. Donald et al, J. Non-Cryst. Solids, l1978;30:77. = 3 cm 20 Bulk glass formation in the Pd-/Ni-/Cu-/Zr- system 21 Recent BMGs with critical size ≥ 10 mm 22 Strength vs. Elastic Limit for Several Classes of Materials 2500 2000 Steels 1500 Titanium Metallic glass alloys 1000 500 Polymers Woods Silica 0 0 1 2 3 Elastic Limit (%) : Metallic Glasses Offer a Unique Combination of High Strength and High Elastic Limit 23 Processing metals as efficiently as plastics: net‐shape forming! Seamaster Planet Ocean Liquidmetal® Limited Edition Superior thermo‐plastic formability : possible to fabricate complex structure without joints Multistep processing can be solved by simple casting Ideal for small expensive IT equipment manufacturing 24 What are Quasicrystals? 25 Crystals can only exhibit certain symmetries In crystals, atoms or atomic clusters repeat periodically, analogous to a tesselation in 2D constructed from a single type of tile. Try tiling the plane with identical units … only certain symmetries are possible A tessellation or tiling of the plane is 26 a collection of plane figures that fills the plane with no overlaps and no gaps. Symmetry axes compatible with periodicity According to the well-known theorems of crystallography, only certain symmetries are allowed: the symmetry of a square, rectangle, parallelogram, triangle or hexagon, but not others, such as pentagons. 27 Crystals can only exhibit certain symmetries Crystals can only exhibit these same rotational symmetries* ..and the symmetries determine many of their physical properties and applications * In 3D, there can be different rotational symmetries along different axes, but they are restricted to the same set (2-, 3-, 4-, and 6-fold) 28 Quasicrystals (Impossible Crystals) were first discovered in the laboratory by Daniel Shechtman, Ilan Blech, Denis Gratias and John Cahn in a beautiful study of an alloy of Al and Mn 29 D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984) 1 m Al6Mn 30 Their surprising claim: “Diffracts electrons like a crystal . But with a symmetry strictly forbidden for crystals” Al6Mn 31 QUASICRYSTALS Similar to crystals, BUT… • Orderly arrangement . But QUASIPERIODIC instead of PERIODIC • Rotational Symmetry . But with FORBIDDEN symmetry • Structure can be reduced to a finite number of repeating units D. Levine and P.J. Steinhardt (1984) Discovery of a Natural Quasicrystal L Bindi, P. Steinhardt, N. Yao and P. Lu Science 324, 1306 (2009) LEFT: Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exterior containsa mixture of spinel, augite, and olivine. The dark material consists predominantly of khatyrkite (CuAl2) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 μm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals.
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